Abstract

We present an analytic resolution of the mass gap problem for four–dimensional Yang–Mills theory based on a dynamic rigidity principle intrinsic to the gauge field evolution. Working in a Hamiltonian framework after gauge fixing, we show that potential low–energy instabilities are confined to a positive spectral sector of the field configuration space. By exploiting the time–integrated structure of the Yang–Mills dynamics, we prove that these unstable compo- nents necessarily induce Laplacian–scale curvature and are therefore absorbed by second–order dissipation. As a consequence, the Yang–Mills Hamiltonian admits a strictly positive spectral lower bound, yielding a non–perturbative mass gap. The argument does not rely on lattice discretiza- tion, large–N limits, or perturbative expansions. Instead, the mass gap emerges from an intrinsic rigidity of the nonlinear gauge dynamics, analogous to rigidity mechanisms governing regularity in incompressible fluid flow. This result identifies a common analytic structure underlying mass gap formation and pro- vides a direct resolution of the mass gap component of the Clay Millennium Yang–Mills problem.